Homework 4: Functions of Several Variables & Probability Solutions

Problem 1: Graph-Function Matching (Optional)

Match the graphs 1-4 to functions i-iv.

Note: This problem requires the visual figure that was referenced but not provided. The matching would involve analyzing: - Function behavior: Identifying linear, quadratic, exponential, or trigonometric patterns - Domain and range: Understanding where functions are defined and their output values - Critical points: Locating maxima, minima, and inflection points - Symmetry properties: Recognizing even/odd functions or periodic behavior

Problem 2: Partial Derivatives

Find the partial derivatives:

\(f(x,y)=3x-2y^2\)
\(f(x,y)=y^7-2x^3+x^2\)
\(f(x,y)=\sin{xy}\)
\(f(x,y)=x \cdot \ln y+\dfrac{x}{y}\)

Problem 3: Directional Derivatives

Compute the directional derivative at the point \((-1,-1)\) along the vector \(\mathbf{v}=[0.6,\,0.8]\):

\(f(x,y) = 3xy\)
\(f(x,y) = e^{x-y}\)

Problem 4: Lake Sevan Navigation

In Lake Sevan, the depth of water at point \((x,y)\) is \(xy^{2}-6x^{2}-3y^{2}\) meters. The ship “Noratus” is at point \((5, 3)\). The first mate suggests sailing north, while the second mate recommends sailing south. Which mate should the captain listen to?

Figure 1: Lake Sevan Depth Visualization

Problem 5: Local Extrema

Does the following function have local extrema? If so, find them:

\(f(x,y) = 3xy\)
\(f(x,y) = x^2-xy\)
\(f(x, y) = 2x^2 - x^3 - y^2\)

You can plot the graph or use the discriminant \(D\) test.

Figure 2: Functions and Their Critical Points

Problem 6: Optimization - Topless Box (Additional)

You have 12 square meters of cardboard and want to make a topless box. What is the maximum volume your box can have?

Hint: Denote height, length and width by \(x\), \(y\) and \(z\). Express \(z\) by \(x\) and \(y\), then express volume by \(x\) and \(y\).

Analytical Solution:
For a square base (x = y), optimal dimension is x = y = 2
Optimal height z = (12 - 4)/(4) = 2
Maximum volume = 2 × 2 × 2 = 8 cubic meters

Problem 7: Convolution (Additional)

For functions \(f(x)=x^2\) and \(g(x) = \begin{cases} 1 & \text{if } x>0\\ 0 & \text{if } x \le 0 \end{cases}\), find \((f*g)(0)\) where: \[(f*g)(x) = \int_{-1}^1 f(y)g(x-y) \, dy\]

Problem 8: Two Dice Probability

Suppose we roll two fair dice. What is the probability of getting:

2 on each of them
at least one 1
exactly one 1
one 1 and one 4
1 on the first die and 4 on the second die

Problem 9: Colored Pencils

There are 2 red, 5 blue and 6 yellow pencils (total: 13). Two pencils are drawn randomly. Find the probability that both are:

red
of the same color
of different colors
not yellow
not green

Problem 10: Dart Throwing

A dart is thrown at a circular target with concentric circles. Circle 1 (innermost) has radius 1m, and each subsequent radius increases by 1m. Find the probability that the dart lands in:

circle 1
a red circle
a yellow circle

Note: Color pattern not specified in the problem, so assuming alternating colors.

Problem 11: Coin Tosses - Odd Heads

A fair coin is tossed 5 times. What is the probability of getting an odd number of heads?

Problem 12: Conditional Probability - Dice

Two fair dice are rolled. What is the probability of getting 1 on at least one die, given that their sum is even?

Problem 13: Playing Cards (Additional)

3 cards are drawn from a deck of 52 cards. What is the probability that the first two cards are queens, and the third one is a diamond ♦?

Problem 14: Reading Books (Additional)

There are 15 books: 5 in Armenian, 10 in French. Ruben cannot read French. If he randomly takes 3 books, what is the probability that he can read at least one?